SOME CHARACTERIZATIONS OF INNER-PRODUCT SPACES BY MAHLON M. DAY 1. Introduction. The theorems of this paper give a number of conditions under which the norm in a real-linear or complex-linear normed space can be defined from an inner product. It should be emphasized that these are not conditions that the space have the same topological structure as an inner- product space but are conditions that the existing norm ||&|| in the space can be defined by the equation ||&||2 = (&, b) from an inner product satisfying the usual axioms. The prototype of all these results is the von Neumann-Jordan condition [11.K1) that a linear metric space B is an inner-product space if and only if ||&+&'||2 + ||&-&'||2 = 2(||i||2+||&'||2) for every pair of points b and b' in B. This implies that a space is an inner-product space if and only if every two- dimensional subspace is an inner-product space; because of this fact most of the computations of this paper can be performed in two-dimensional spaces. In a real two-dimensional space it is well known that the space is an inner- product space if and only if the set of points of norm one is an ellipse ; to prove sufficiency of our conditions in the real two-dimensional case it turns out to be convenient to have means of associating a specific ellipse with any given unit sphere and then to prove that, under the given conditions, the unit sphere is that particular ellipse. Two ways are used to select such an ellipse for several of these proofs. (A) By Theorem 1 of the preceding paper [5], if C is the set of points of norm one, there is a parallelogram P circumscribed about C so that the mid- point of each side of P is on C. The largest ellipse E inscribed in P also meets the midpoints of the sides of P. Under the hypotheses of Theorems 2.1 and 3.1 it can be shown that C is E. (B) Loewner has shown (in some unpublished work) that there exists a unique ellipse E' of minimal area circumscribed about C, and that this ellipse touches C in at least four points. Under the conditions of Theorems 2.1 and 4.1, it is possible to show that C is E'. It turns out that the real-linear cases of the following theorems can be proved more easily than the complex-linear cases; this is due to the avail- ability of the geometrical devices (A) and (B) in the real two-dimensional case. The proofs for the real-linear cases are given in §§2 through 6. In §7 the complex cases of these theorems are derived from the real cases Presented to the Society, April 29, 1944; received by the editors September 27, 1946. (') Numbers in brackets refer to the bibliography at the end of the paper. 320 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
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SOME CHARACTERIZATIONS OFINNER-PRODUCT SPACES
BY
MAHLON M. DAY
1. Introduction. The theorems of this paper give a number of conditions
under which the norm in a real-linear or complex-linear normed space can be
defined from an inner product. It should be emphasized that these are not
conditions that the space have the same topological structure as an inner-
product space but are conditions that the existing norm ||&|| in the space
can be defined by the equation ||&||2 = (&, b) from an inner product satisfying
the usual axioms.
The prototype of all these results is the von Neumann-Jordan condition
[11.K1) that a linear metric space B is an inner-product space if and only if
||&+&'||2 + ||&-&'||2 = 2(||i||2+||&'||2) for every pair of points b and b' in B.
This implies that a space is an inner-product space if and only if every two-
dimensional subspace is an inner-product space; because of this fact most of
the computations of this paper can be performed in two-dimensional spaces.
In a real two-dimensional space it is well known that the space is an inner-
product space if and only if the set of points of norm one is an ellipse ; to prove
sufficiency of our conditions in the real two-dimensional case it turns out to
be convenient to have means of associating a specific ellipse with any given
unit sphere and then to prove that, under the given conditions, the unit
sphere is that particular ellipse. Two ways are used to select such an ellipse
for several of these proofs.
(A) By Theorem 1 of the preceding paper [5], if C is the set of points of
norm one, there is a parallelogram P circumscribed about C so that the mid-
point of each side of P is on C. The largest ellipse E inscribed in P also meets
the midpoints of the sides of P. Under the hypotheses of Theorems 2.1 and
3.1 it can be shown that C is E.
(B) Loewner has shown (in some unpublished work) that there exists a
unique ellipse E' of minimal area circumscribed about C, and that this
ellipse touches C in at least four points. Under the conditions of Theorems
2.1 and 4.1, it is possible to show that C is E'.
It turns out that the real-linear cases of the following theorems can be
proved more easily than the complex-linear cases; this is due to the avail-
ability of the geometrical devices (A) and (B) in the real two-dimensional
case. The proofs for the real-linear cases are given in §§2 through 6.
In §7 the complex cases of these theorems are derived from the real cases
Presented to the Society, April 29, 1944; received by the editors September 27, 1946.
(') Numbers in brackets refer to the bibliography at the end of the paper.
320
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INNER-PRODUCT SPACES 321
and from certain elementary but interesting relationships between a complex-
linear space B and the real-linear space A associated with it by forgetting
to use the non-real scalar multiples. For example, Theorem 7.1 says that
there is a natural, one-to-one, real-linear, norm-preserving mapping between
all of A*, the space of real-valued, real-linear functions on A, and all of B*,
the space of complex-valued, complex-linear functions on B. Theorem 7.2
says that the norm in a complex-linear normed space B arises from a complex
inner product if and only if it arises from a real inner product. These results
easily give the complex cases of the theorems 1 to 7 from the real cases which
are to be proved in §§2 to 6.
The first of the criteria to be given is a formal weakening of the condition
given by Jordan and von Neumann in complex-linear metric spaces. If the
space is assumed to be normed, it is obvious that one of the points &and b'
used in the test condition can be chosen to be of norm one. The first theorem
of this paper states that the additional restriction that both points are of
norm one can be imposed (Theorem 2.1).
Further theorems give characterizations in terms of: a condition using
points both of B and its conjugate space (Theorem 3.1), uniform convexity
(Theorem 4.1), various types of orthogonality (Theorems 5.1 and 5.2) (com-
pleting certain results of James [9]), and normality (extending a result of
G. Birkhoff [2]).2. A weakening of the von Neumann-Jordan condition. We begin with
the definition of a real inner product space ; we shall be concerned in the next
five sections only with the real linear case; the complex case and its relation
to the real is given in §7. A (real) inner product in a real-linear(2) space B is
a real-valued function of two variables (b, b') in B satisfying the conditions:
Rl. (b, b)>0iib9*0;
R2. (b,b')=(b',b);R3. (mibi-\-mibi, bz)=mi(bi, b¿)-\-mi(bi, bs) for all elements &,- of B and
real numbers w¿.
A real-linear normed space 23 is a real inner-product space if there exists a
real inner product such that ||&||2 = (Z>, b) for every b in B.
Theorem 2A. A normed space B is an inner-product space if and only if it
satisfies the condition
(*) ||6i + &2||2 + ||ô, -&2||2 = 4 #N1-NI "I-
It is well known that a two-dimensional normed space is an inner-product
space if and only if the unit sphere in B is an ellipse; from this we see that a
special case of 2.1 is the following theorem.
(a) A linear space is an Abelian group B (written additively) in which multiplication by
scalars is defined to satisfy certain associative and distributive laws (see Banach [l]). If the
scalars are real (complex) numbers, B will be called a real-linear (complex-linear) space.
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322 M. M. DAY [September
Theorem 2.2. If B is a two-dimensional real-linear space and C is the set
of points of norm one, C is an ellipse if and only if (*) holds.
Two methods of proof are available for this theorem. One begins with the
circumscribed ellipse of smallest area and proves that it is C. A second proof
using the method (A) of the introduction was discovered earlier but takes
several times as much space; it will be outlined at the end of this section. The
first proof uses some lemmas due to Loewner.
Lemma 2.3 (Loewner). // C is a symmetric closed convex curve, there exists
a unique ellipse of minimal area circumscribed about C.
Let Ei and E2 be ellipses of minimal area circumscribed about C, make an
affine transformation reducing both ellipses to principal axes so that equations
for them may be written in the form x2+y2 = l, ax2+by2 = l. Since these are
of equal area, it, ab = \. The condition that they contain C shows that
*2+y2^l and ax2+by2^l for (x, y) on C. Hence (a+l)*2/2+(& + l)y2/2^1
also for points of C; that is, the ellipse (a + í)x2/2-\-(b + í)y2/2 = 1 contains C.
The area of this last ellipse is 27r/((a + l)(è + l))1/2. This is less thamr unless
a =5 = 1 so the minimal ellipse is unique.
Lemma 2.4 (Loewner). The minimal ellipse circumscribed about a sym-
metric closed convex C touches C in at least four points.
If the minimal ellipse E is written in the form #2+y2 = l, where (1, 0) is a
point of contact of C and E, then the ellipses E, with equations x2/(l+e)
+ (l+e)y2=l also have the minimal area so do not contain C. For each
e>0 let pf be a point of C not in Ee; then there is a sequence e„—»0 such that
pCn converges to some point p. Since all pt are on C, p is on C; moreover p,
is within é of E, so p is also on E. Now the intersections of E and E, have y
coordinates +l/(2+é)1/2; hence the absolute value of the y coordinate of
p is at least 1/21/2. Therefore p is not either point (±1, 0) and is on both
C and E. By symmetry there are at least four points of contact of C and E.
To apply Lemma 2.4 to the proof of Theorem 2.2, let C be the set of points
b for which ||Z>|| =1 and let E be the ellipse of minimal area circumscribed
about C. Also define a new norm \b\ in F so that E is the set of all those b for
which |ô| =1. Since \b\ can be defined from an inner product, the relation
(*) holds for \b\. Since C is inside E, \\b\\ ̂ \b\ for every b. Now let x and
y^+x be two points of contact of C and E. Then 4= |x+;y| 2+|x — y|2
= |k+;y||2 + ||:e — ;y||2=4; hence |]*+;y|| = |*+;y| and ||:s — y|| = \x — y\. That
is to say, if C and E meet at two points, then they meet at two more points
roughly half way between the original points.
Repeating the process by induction shows that C and E agree on a dense
set of points; hence C is E. 2.2 now implies 2.1 by the von Neumann-Jordan
condition.
The second proof of Theorem 2.2 is too long to give here in full but will be
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1947] INNER-PRODUCT SPACES 323
sketched instead. By Theorem 1 of my preceding paper [5] there exists a
parallelogram P circumscribed about C so that the midpoint of each side of P
is on C. Make an affine transformation of B that carries P into a square and
draw the inscribed circle E of P. We wish to show that when (*) holds, C is
E. Represent C by the polar coordinate equation r=f(6) where the polar
coordinate system is so chosen that 0=0 is a line from the origin to the mid-
point of one side of P. The major part of the proof is the proof of the follow-
ing lemma.
Lemma 2.5. If (*) holds, if 0 ̂ 0o<0i air/2, and if f( ±00) = 1 =/(±6i), then/((00+00/2) =/(- (00+00/2) =/(tt/2 - (00+00/2) =/( -tt/2 + (0o+0O/2) = 1.
To prove this lemma let </> = (l, 0O), ^ = (1, 00, * = (!> -0o), and ^
= (1, -di). Define h= \<¡>-\-^\ /2 = 13>+*|/2 =cos (0i-0o)/2 and ki
= |«^.-^|/2=| 4>-^|/2=cos [1r/2-(01-0o)/2]. Then let r1=/((0o+0O/2),
r,=/(r/2-(00+00/2), 2?i=/(-(0o+0i)/2), and P2=/(-7r/2+(0o+0O/2).
Then ||*+*||/2 =*,/>i, ||*-*||/2 = *,/&. \\$+*\\/2=ki/Ri, and ||*-*||/2= k2/ri.
Dividing (*) by 4 and substituting from the equations just above gives
two conditions on these variables:
(1) k\/r\ + A2/P2 = 1,
(2) kl/Rl + kl/rl = 1.
Let 0' = (00+00/2, <p' = (fi, 0'), and $' = (Ru -B'). The horizontal projection
of <j>' + & is rx cos 0'+2?i cos (-$')= (n+Ri) cos 0'. Since C is inside P,
||c6' + i>'|| â(/i+2?i) cos 0'. A similar argument with the vertical coordinates
shows that \\<f>'- $'|| ^ (ri+Ri) sinö'. Similar arguments with \p' = (r2, tt/2-0')
and *' = (2?2, B'-v/2) give ||^'+¥'|| =(r2+P0 cos (ir/2-0') and ||^/-'4r,||
è(r2+P2) sin (x/2-00. Substituting in (*) gives (/i+2202g4 and (r2+P2)2
^4 or
(3) n + 2?j ̂ 2,
(4) r2 + 2?2 ^ 2.
It can be shown by straightforward computation that the only solution of
these four conditions is n = r2 = 2?i = 2?2 = 1 ; to prove this it suffices by the
symmetry of the equations to show that no solution exists with r!>l. This
shows that the points c/>', \p', <£', and W are on E as well as on C.
This lemma says that if C and E touch in two symmetric pairs of points,
then they touch at some intermediate symmetric pairs of points. Starting
with 0o = O, 01=7r/2, and applying this lemma repeatedly shows that C and
E coincide on a dense set ; hence C is E.
3. A characterization by an equation involving points of both B and its
conjugate space. In the paper [6] I proved the relation that if |[&.-|| =||ß,||
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324 M. M. DAY [September
= ßi(b<) = lior i=l, 2, then 8è||ft+ft|| -¡¡bi + hW +l|ft-ft|| M^-^W =4-The proof is trivial for
í-(íi+í*)/||ii+*í||, s' = (si+si)/\\si+sí\\, ff = (<ri+«ri)/1ki+cr,||, <*»"* «'= (oí +02 )/\\o{ +<rí II. Then s, s', a, and o' are related as are Si, s', cr,-, and ai
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326 M. M. DAY [September
for í = 1 or 2; that is, (a') ||j|| =||<r|| =<r(s) = 1 =||5'|| =||<r'|| =(r'(s'), and